Observe that \(\Vert  X - \mathbb {E}[X\mid Y] \Vert _{k}\le 2 \min \left( (k!)^{1/(2k)}, \Vert X\Vert _k\right)\)  (see [1]}), so applying the triangle inequality in (REF ) we obtain
\(\left\Vert  f^{(r-1)}(Y) \right\Vert _2 &\le \sum _{\lambda \in \Pi _r} c_{\lambda } \left\Vert  g^{\lambda }(Y) \right\Vert _2 \le 2^rC_r ~ \min \left( \gamma _r, \Vert X\Vert _{2rq_r}^r \right),\) 
